Ch 13. 1: Cournot Oligopoly

Via worked example.

In this section I will introduce the Cournot oligopoly model using the following demand curve, P=63-Q with marginal costs=3.

Cournot models assume the firms will choose quantities and then the prices adjust. It’s helpful to write out a generic inverse demand curve such as P=A-Q and to assume constant marginal costs of C. Firm 1 produces Q1 and firm 2 produces Q2. We assume Q= Q1 + Q2 and note that the market price is determined by the independent quantity setting choices of the two firms. The larger the quantity, the lower the price. The market will only bear so many units. If your rival brings a larger quantity to the market, you’ll have an incentive to bring a smaller quantity and vice versa. Ideally both would collude by producing 1/2 of the monopoly quantity, but this often doesn’t happen because exactly when your rival is producing 1/2 the monopoly quantity you can strictly gain (increase profits) by increasing your own output!

Importantly, in standard Cournot models we assume that the firms are making their choices simultaneously. As in game theory the term ‘simultaneous’ actually refers to information available to each firm when choosing and not to literal chronological time. Simultaneous timing means that the firm does not observe the choice of their rivals until after they make their own selection. If one firm is able to observe the rival’s choice we have a sequential game and typically refer to this as a Stackelberg oligopoly (with a clearly defined leader firm and another follower firm).

We could certainly build out the Cournot model using the generic demand and costs mentioned above, but I think it’s probably more effective to do this in the context of a numerical example. As I mentioned in the introduction we’ll use the following linear demand curve: P=63-Q. Assume we have two firms and both have the same costs, MC=3. Because we are modeling this as a Cournot interaction we are assuming both firms are going to chose their own quantities strategically. Therefore I will start by writing down the profit maximization problem. Also as a consequence of Cournot modeling we are assuming the firms make their choices simultaneously. Again this is not a statement about chronological time so much as it’s importantly a statement about information: Firms choose without knowing the choice of their rival.

Here’s firm 1’s problem beginning with an expression for profits, then expanding algebraically:

\(\huge \substack{ \pi_1 = (63 - Q_1 - Q_2)Q_1 - 3Q_1 \\\\\\\\\ \pi_1 = 63Q_1 - Q_1^{2} - Q_2Q_1 - 3Q_1 \\\\\\\\\\ \pi_1 = 60Q_1 - Q_1^{2} - Q_2Q_1 }\)

Because the firm’s choice is to find their optimal quantity, we need to take the partial derivative with respect to Q1, then solve for Q1:

\(\huge \substack{ \frac{\partial \pi_1}{\partial Q_1} = 60 - 2Q_1 - Q_2=0\\\\\\\ Q_1 = 30 - \frac{1}{2}Q_2 }\)

This last line is firm 1’s reaction curve which gives its optimal choice, Q1, as a function of the rival’s choice, Q2. Notice that I multiplied through by 1/2 rather than write as a large fraction? This is to simplify solving the system of reaction curves in the future. As a matter of fact, any time the term a-c (from P=a - q & MC=c) is even or ‘nicely divisible by 2’ I recommend doing it this way. In this case A=63 and c=3 so A-C = 60 which is even. Let’s set this aside for a second. We need to get firm 2’s reaction curve.

Now this particular problem has a sort of symmetry. The demand curve is going to treat units produced by firm 1 interchangably with units produced by firm 2. We know this because the coefficients in the demand curve are the same for both Q1 and Q2, from: P=60 - Q1 - Q2. Also, they have the same marginal costs, 3. This means we could just flip the labels to arrive at Q2 = 30 -(1/2)Q1. But at least once it’s good to prove to ourselves this works:

Here’s firm 2’s problem:

\(\huge \substack{ \pi_2 = (63 - Q_2 - Q_2)Q_2 - 3Q_2 \\\\\\\\\ \pi_2 = 63Q_2 - Q_2^{2} - Q_2Q_1 - 3Q_2 \\\\\\\\\\ \pi_2 = 60Q_2 - Q_2^{2} - Q_2Q_1 }\)

Because the firm’s choice is to find their optimal quantity, we need to take the partial derivative with respect to Q2, then solve for Q2:

\(\huge \substack{ \frac{\partial \pi_2}{\partial Q_2} = 60 - Q_1 - 2Q_2=0\\\\\\\ Q_2 = 30 - \frac{1}{2}Q_1 }\)

Now we have both reaction curves. These are not quite our solutions. Both functions just give us the firm’s optimal choice given their rivals choice. For example if Q2=0, then firm 1 is a monopoly and wants to set Q1 = 30. But if firm 2 realizes this, it best responds by selecting: Q2(30) = 30 - (1/2)(30) = 15. But then if firm 1 realizes the, it best responds by selecting: Q1(15) = 30 - (1/2)(15) = 22.5. And so on. This will eventually converge to the equilibrium but there’s an easier way. We want to find the specific equilibrium where no one wants to change their selection from the outset. We have two equations with two unknowns and can solve by substitution to find the optimal Q1* and Q2*.

Let’s substitute Q2(Q1) where Q2 appears in firm 1’s reaction curve. This gives us an equation with a single unknown:

\(\huge \substack{ Q_1 = 30 - \frac{1}{2}(30 - \frac{1}{2}Q_1) \\\\\\ Q_1 = 30 - 15 +\frac{1}{4}Q_1 \\\\\\\ \frac{3}{4}Q_1 = 15 \\\\\\\\ Q_1 = 20 }\)

From the above collection of steps please notice that I’ve got a term, 3/4 times Q1. This will happen every time. If you reach that point in the calculation and instead find the term 5/4 times Q1 you know you’ve made a sign error (Likely wrote -1/2 * -1/2Q1 as -1/2Q1). This is a good tip to help troubleshoot! Once you’ve got the numerical result it’s a good idea to test that it is indeed an equilibrium by plugging into both reaction curves separately:

\(\huge \substack{Q_1(20) = 30 - \frac{1}{2}(20) = 20\\\\\\\\\ Q_2(20) = 30 - \frac{1}{2}(20) = 20}\)

Lastly we need to find the market price: P = 63 - (20) -(20) = 23

And then the profits for both firms: Profits = (23 - 3)(20) = 400 to each firm.